Proof and proving in school mathematics instruction: Making the elementary grades part of the equation

Stylianides, A.J.

PhD Thesis, University of Michigan, Ann Arbor, 2005

The concept of proof and the practice of proving have not typically been a focus of attention by those involved in the teaching and learning of mathematics in the elementary grades. This constitutes a serious lack in the way we conceptualize elementary school mathematics. Because proof and proving are at the core of doing and knowing mathematics, we cannot have a viable elementary school mathematics curriculum, or opportunities for students to learn it that have integrity, without having a way to incorporate proof and proving into a coherent conception of mathematics instruction in the early grades. This study investigates (1) how proof and proving might be conceptualized in the context of elementary mathematics instruction, and (2) how this conceptualization can inform the work that elementary teachers would need to do, and the knowledge that they would need to have, to promote proof and proving in their classrooms.
The study is structured around three interrelated strands of work. The first strand illuminates the nature of proof in the early grades by identifying and exploring parameters potentially determinant of which arguments qualify as proofs. The second strand capitalizes on the first and sets forth a framework about the meaning of proof in K-12 mathematics instruction. Additionally, it uses this framework as a tool to examine instruction in the early grades in order to clarify aspects of teachers' role in fostering proof and proving. As a result, the second strand also develops a framework about instructional practices for cultivating proof and proving that make sense even in the early grades. The third strand investigates the knowledge of proof needed for cultivating proof and proving in elementary mathematics instruction. It advances a classification of different kinds of knowledge of proof that elementary teachers might need, paying particular attention to what is involved for teachers as they mobilize opportunities for their students to engage in proving. The study pursues the three strands of work both conceptually, using scholarly work on proof (including work on mathematical practice), and empirically, using data from the teaching practice of an elementary teacher who was trying to cultivate her students' reasoning skills.
The products of the study offer insight into what it might mean, and what it would take, to make proof and proving central to elementary children's learning of mathematics. The conceptual analytic tools developed by the study contribute to theory building in the teaching and learning of proof and proving in the early grades but also more broadly. Furthermore, these tools can support the design of materials for the professional education of elementary teachers, and provide guidance for mathematics teacher educators who might implement them